论文标题
$ c^1 $ petrov-galerkin方法和高斯搭配方法
A $C^1$ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence
论文作者
论文摘要
在本文中，我们介绍和研究$ c^1 $ PETROV-GALERKIN和GAUSS搭配方法，具有任意多项式$ K $（$ \ ge 3 $）的一维椭圆方程的方法。我们证明，解决方案及其衍生近似值在所有网格点都以$ 2K-2 $的速率收敛；解决方案的近似值在每个元素的特殊雅各比多项式的所有内部根部都是超对面作为副产品，我们证明Petrov-Galerkin解决方案和高斯搭配解决方案都超过了$ H^2 $，$ H^1 $和$ l^2 $ NORMS中精确解决方案的特定雅各比投影。所有理论发现均通过数值实验证实。
In this paper, we present and study $C^1$ Petrov-Galerkin and Gauss collocation methods with arbitrary polynomial degree $k$ ($\ge 3$) for one-dimensional elliptic equations. We prove that, the solution and its derivative approximations converge with rate $2k-2$ at all grid points; and the solution approximation is superconvergent at all interior roots of a special Jacobi polynomial of degree $k+1$ in each element, the first-order derivative approximation is superconvergent at all interior $k-2$ Lobatto points, and the second-order derivative approximation is superconvergent at $k-1$ Gauss points, with an order of $k+2$, $k+1$, and $k$, respectively. As a by-product, we prove that both the Petrov-Galerkin solution and the Gauss collocation solution are superconvergent towards a particular Jacobi projection of the exact solution in $H^2$, $H^1$, and $L^2$ norms. All theoretical findings are confirmed by numerical experiments.
